xenial (1) greenspline.1gmt.gz
NAME
greenspline - Interpolate using Green's functions for splines in 1-3 dimensions
SYNOPSIS
greenspline [ table ] [ [1|2|3|4|5,]gradfile ] [ [n|v]cut[/file] ] [ mode ] [ grdfile ] [
xinc[/yinc[/zinc]] ] [ ] [ nodefile ] [ az|x/y/z ] [ west/east/south/north[/zmin/zmax][r] ] [
c|t|l|r|p|q[pars] ] [ maskgrid ] [ [level] ] [ ] [ -b<binary> ] [ -d<nodata> ] [ -f<flags> ] [
-h<headers> ] [ -o<flags> ] [ -x[[-]n] ] [ -:[i|o] ]
Note: No space is allowed between the option flag and the associated arguments.
DESCRIPTION
greenspline uses the Green's function G(x; x') for the chosen spline and geometry to interpolate data at
regular [or arbitrary] output locations. Mathematically, the solution is composed as w(x) = sum {c(i)
G(x'; x(i))}, for i = 1, n, the number of data points {x(i), w(i)}. Once the n coefficients c(i) have
been found the sum can be evaluated at any output point x. Choose between minimum curvature, regularized,
or continuous curvature splines in tension for either 1-D, 2-D, or 3-D Cartesian coordinates or spherical
surface coordinates. After first removing a linear or planar trend (Cartesian geometries) or mean value
(spherical surface) and normalizing these residuals, the least-squares matrix solution for the spline
coefficients c(i) is found by solving the n by n linear system w(j) = sum-over-i {c(i) G(x(j); x(i))},
for j = 1, n; this solution yields an exact interpolation of the supplied data points. Alternatively,
you may choose to perform a singular value decomposition (SVD) and eliminate the contribution from the
smallest eigenvalues; this approach yields an approximate solution. Trends and scales are restored when
evaluating the output.
REQUIRED ARGUMENTS
None.
OPTIONAL ARGUMENTS
table The name of one or more ASCII [or binary, see -bi] files holding the x, w data points. If no file
is given then we read standard input instead.
-A[1|2|3|4|5,]gradfile
The solution will partly be constrained by surface gradients v = v*n, where v is the gradient
magnitude and n its unit vector direction. The gradient direction may be specified either by
Cartesian components (either unit vector n and magnitude v separately or gradient components v
directly) or angles w.r.t. the coordinate axes. Specify one of five input formats: 0: For 1-D data
there is no direction, just gradient magnitude (slope) so the input format is x, gradient. Options
1-2 are for 2-D data sets: 1: records contain x, y, azimuth, gradient (azimuth in degrees is
measured clockwise from the vertical (north) [Default]). 2: records contain x, y, gradient,
azimuth (azimuth in degrees is measured clockwise from the vertical (north)). Options 3-5 are for
either 2-D or 3-D data: 3: records contain x, direction(s), v (direction(s) in degrees are
measured counter-clockwise from the horizontal (and for 3-D the vertical axis). 4: records contain
x, v. 5: records contain x, n, v. Append name of ASCII file with the surface gradients (following
a comma if a format is specified).
-C[n|v]cut[/file]
Find an approximate surface fit: Solve the linear system for the spline coefficients by SVD and
eliminate the contribution from all eigenvalues whose ratio to the largest eigenvalue is less than
cut [Default uses Gauss-Jordan elimination to solve the linear system and fit the data exactly].
Optionally, append /file to save the eigenvalue ratios to the specified file for further analysis.
Finally, if a negative cut is given then /file is required and execution will stop after saving
the eigenvalues, i.e., no surface output is produced. Specify -Cv to use the largest eigenvalues
needed to explain cut % of the data variance. Alternatively, use -Cn to select the cut largest
eigenvalues. If a file is given with -Cv then we save the eigenvalues instead of the ratios.
-Dmode Sets the distance flag that determines how we calculate distances between data points. Select mode
0 for Cartesian 1-D spline interpolation: -D0 means (x) in user units, Cartesian distances, Select
mode 1-3 for Cartesian 2-D surface spline interpolation: -D1 means (x,y) in user units, Cartesian
distances, -D2 for (x,y) in degrees, Flat Earth distances, and -D3 for (x,y) in degrees, Spherical
distances in km. Then, if PROJ_ELLIPSOID is spherical, we compute great circle arcs, otherwise
geodesics. Option mode = 4 applies to spherical surface spline interpolation only: -D4 for (x,y)
in degrees, use cosine of great circle (or geodesic) arcs. Select mode 5 for Cartesian 3-D surface
spline interpolation: -D5 means (x,y,z) in user units, Cartesian distances.
-Ggrdfile
Name of resulting output file. (1) If options -R, -I, and possibly -r are set we produce an
equidistant output table. This will be written to stdout unless -G is specified. Note: for 2-D
grids the -G option is required. (2) If option -T is selected then -G is required and the output
file is a 2-D binary grid file. Applies to 2-D interpolation only. (3) If -N is selected then the
output is an ASCII (or binary; see -bo) table; if -G is not given then this table is written to
standard output. Ignored if -C or -C0 is given.
-Ixinc[/yinc[/zinc]]
Specify equidistant sampling intervals, on for each dimension, separated by slashes.
-L Do not remove a linear (1-D) or planer (2-D) trend when -D selects mode 0-3 [For those Cartesian
cases a least-squares line or plane is modeled and removed, then restored after fitting a spline
to the residuals]. However, in mixed cases with both data values and gradients, or for spherical
surface data, only the mean data value is removed (and later and restored).
-Nnodefile
ASCII file with coordinates of desired output locations x in the first column(s). The resulting w
values are appended to each record and written to the file given in -G [or stdout if not
specified]; see -bo for binary output instead. This option eliminates the need to specify options
-R, -I, and -r.
-Qaz|x/y/z
Rather than evaluate the surface, take the directional derivative in the az azimuth and return the
magnitude of this derivative instead. For 3-D interpolation, specify the three components of the
desired vector direction (the vector will be normalized before use).
-Rxmin/xmax[/ymin/ymax[/zminzmax]]
Specify the domain for an equidistant lattice where output predictions are required. Requires -I
and optionally -r.
1-D: Give xmin/xmax, the minimum and maximum x coordinates.
2-D: Give xmin/xmax/ymin/ymax, the minimum and maximum x and y coordinates. These may be Cartesian
or geographical. If geographical, then west, east, south, and north specify the Region of
interest, and you may specify them in decimal degrees or in [+-]dd:mm[:ss.xxx][W|E|S|N] format.
The two shorthands -Rg and -Rd stand for global domain (0/360 and -180/+180 in longitude
respectively, with -90/+90 in latitude).
3-D: Give xmin/xmax/ymin/ymax/zmin/zmax, the minimum and maximum x, y and z coordinates. See the
2-D section if your horizontal coordinates are geographical; note the shorthands -Rg and -Rd
cannot be used if a 3-D domain is specified.
-Sc|t|l|r|p|q[pars]
Select one of six different splines. The first two are used for 1-D, 2-D, or 3-D Cartesian splines
(see -D for discussion). Note that all tension values are expected to be normalized tension in the
range 0 < t < 1: (c) Minimum curvature spline [Sandwell, 1987], (t) Continuous curvature spline in
tension [Wessel and Bercovici, 1998]; append tension[/scale] with tension in the 0-1 range and
optionally supply a length scale [Default is the average grid spacing]. The next is a 1-D or 2-D
spline: (l) Linear (1-D) or Bilinear (2-D) spline; these produce output that do not exceed the
range of the given data. The next is a 2-D or 3-D spline: (r) Regularized spline in tension
[Mitasova and Mitas, 1993]; again, append tension and optional scale. The last two are spherical
surface splines and both imply -D4: (p) Minimum curvature spline [Parker, 1994], (q) Continuous
curvature spline in tension [Wessel and Becker, 2008]; append tension. The G(x'; x') for the last
method is slower to compute (a series solution) so we pre-calculate values and use cubic spline
interpolation lookup instead. Optionally append +nN (an odd integer) to change how many points to
use in the spline setup [10001]. The finite Legendre sum has a truncation error [1e-6]; you can
lower that by appending +elimit at the expense of longer run-time.
-Tmaskgrid
For 2-D interpolation only. Only evaluate the solution at the nodes in the maskgrid that are not
equal to NaN. This option eliminates the need to specify options -R, -I, and -r.
-V[level] (more ...)
Select verbosity level [c].
-W Expect data weights in the final input column, typically given as weight = 1 / sigma, the data
uncertainty. This results in a weighted least squares fit. Note that this only has an effect if
-CC is used.
-bi[ncols][t] (more ...)
Select native binary input. [Default is 2-4 input columns (x,w); the number depends on the chosen
dimension].
-bo[ncols][type] (more ...)
Select native binary output.
-d[i|o]nodata (more ...)
Replace input columns that equal nodata with NaN and do the reverse on output.
-f[i|o]colinfo (more ...)
Specify data types of input and/or output columns.
-h[i|o][n][+c][+d][+rremark][+rtitle] (more ...)
Skip or produce header record(s).
-icols[l][sscale][ooffset][,...] (more ...)
Select input columns (0 is first column).
-ocols[,...] (more ...)
Select output columns (0 is first column).
-r (more ...)
Set pixel node registration [gridline].
-x[[-]n] (more ...)
Limit number of cores used in multi-threaded algorithms (OpenMP required).
-^ or just -
Print a short message about the syntax of the command, then exits (NOTE: on Windows use just -).
-+ or just +
Print an extensive usage (help) message, including the explanation of any module-specific option
(but not the GMT common options), then exits.
-? or no arguments
Print a complete usage (help) message, including the explanation of options, then exits.
--version
Print GMT version and exit.
--show-datadir
Print full path to GMT share directory and exit.
1-D EXAMPLES
To resample the x,y Gaussian random data created by gmtmath and stored in 1D.txt, requesting output every
0.1 step from 0 to 10, and using a minimum cubic spline, try
gmt math -T0/10/1 0 1 NRAND = 1D.txt
gmt psxy -R0/10/-5/5 -JX6i/3i -B2f1/1 -Sc0.1 -Gblack 1D.txt -K > 1D.ps
gmt greenspline 1D.txt -R0/10 -I0.1 -Sc -V | psxy -R -J -O -Wthin >> 1D.ps
To apply a spline in tension instead, using a tension of 0.7, try
gmt psxy -R0/10/-5/5 -JX6i/3i -B2f1/1 -Sc0.1 -Gblack 1D.txt -K > 1Dt.ps
gmt greenspline 1D.txt -R0/10 -I0.1 -St0.7 -V | psxy -R -J -O -Wthin >> 1Dt.ps
2-D EXAMPLES
To make a uniform grid using the minimum curvature spline for the same Cartesian data set from Davis
(1986) that is used in the GMT Technical Reference and Cookbook example 16, try
gmt greenspline table_5.11 -R0/6.5/-0.2/6.5 -I0.1 -Sc -V -D1 -GS1987.nc
gmt psxy -R0/6.5/-0.2/6.5 -JX6i -B2f1 -Sc0.1 -Gblack table_5.11 -K > 2D.ps
gmt grdcontour -JX6i -B2f1 -O -C25 -A50 S1987.nc >> 2D.ps
To use Cartesian splines in tension but only evaluate the solution where the input mask grid is not NaN,
try
gmt greenspline table_5.11 -Tmask.nc -St0.5 -V -D1 -GWB1998.nc
To use Cartesian generalized splines in tension and return the magnitude of the surface slope in the NW
direction, try
gmt greenspline table_5.11 -R0/6.5/-0.2/6.5 -I0.1 -Sr0.95 -V -D1 -Q-45 -Gslopes.nc
Finally, to use Cartesian minimum curvature splines in recovering a surface where the input data is a
single surface value (pt.d) and the remaining constraints specify only the surface slope and direction
(slopes.d), use
gmt greenspline pt.d -R-3.2/3.2/-3.2/3.2 -I0.1 -Sc -V -D1 -A1,slopes.d -Gslopes.nc
3-D EXAMPLES
To create a uniform 3-D Cartesian grid table based on the data in table_5.23 in Davis (1986) that
contains x,y,z locations and a measure of uranium oxide concentrations (in percent), try
gmt greenspline table_5.23 -R5/40/-5/10/5/16 -I0.25 -Sr0.85 -V -D5 -G3D_UO2.txt
2-D SPHERICAL SURFACE EXAMPLES
To recreate Parker's [1994] example on a global 1x1 degree grid, assuming the data are in file
mag_obs_1990.d, try
greenspline -V -Rg -Sp -D3 -I1 -GP1994.nc mag_obs_1990.d
To do the same problem but applying tension of 0.85, use
greenspline -V -Rg -Sq0.85 -D3 -I1 -GWB2008.nc mag_obs_1990.d
CONSIDERATIONS
1. For the Cartesian cases we use the free-space Green functions, hence no boundary conditions are
applied at the edges of the specified domain. For most applications this is fine as the region
typically is arbitrarily set to reflect the extent of your data. However, if your application requires
particular boundary conditions then you may consider using surface instead.
2. In all cases, the solution is obtained by inverting a n x n double precision matrix for the Green
function coefficients, where n is the number of data constraints. Hence, your computer's memory may
place restrictions on how large data sets you can process with greenspline <greenspline.html>.
Pre-processing your data with blockmean <blockmean.html>, blockmedian <blockmedian.html>, or blockmode
<blockmode.html> is recommended to avoid aliasing and may also control the size of n. For information,
if n = 1024 then only 8 Mb memory is needed, but for n = 10240 we need 800 Mb. Note that greenspline
<greenspline.html> is fully 64-bit compliant if compiled as such. For spherical data you may consider
decimating using gmtspatial <gmtspatial.html> nearest neighbor reduction.
3. The inversion for coefficients can become numerically unstable when data neighbors are very close
compared to the overall span of the data. You can remedy this by pre-processing the data, e.g., by
averaging closely spaced neighbors. Alternatively, you can improve stability by using the SVD solution
and discard information associated with the smallest eigenvalues (see -C).
4. The series solution implemented for -Sq was developed by Robert L. Parker, Scripps Institution of
Oceanography, which we gratefully acknowledge.
5. If you need to fit a certain 1-D spline through your data points you may wish to consider sample1d
<sample1d.html> instead. It will offer traditional splines with standard boundary conditions (such as
the natural cubic spline, which sets the curvatures at the ends to zero). In contrast, greenspline's
1-D spline, as is explained in note 1, does not specify boundary conditions at the end of the data
domain.
TENSION
Tension is generally used to suppress spurious oscillations caused by the minimum curvature requirement,
in particular when rapid gradient changes are present in the data. The proper amount of tension can only
be determined by experimentation. Generally, very smooth data (such as potential fields) do not require
much, if any tension, while rougher data (such as topography) will typically interpolate better with
moderate tension. Make sure you try a range of values before choosing your final result. Note: the
regularized spline in tension is only stable for a finite range of scale values; you must experiment to
find the valid range and a useful setting. For more information on tension see the references below.
REFERENCES
Davis, J. C., 1986, Statistics and Data Analysis in Geology, 2nd Edition, 646 pp., Wiley, New York,
Mitasova, H., and L. Mitas, 1993, Interpolation by regularized spline with tension: I. Theory and
implementation, Math. Geol., 25, 641-655.
Parker, R. L., 1994, Geophysical Inverse Theory, 386 pp., Princeton Univ. Press, Princeton, N.J.
Sandwell, D. T., 1987, Biharmonic spline interpolation of Geos-3 and Seasat altimeter data, Geophys. Res.
Lett., 14, 139-142.
Wessel, P., and D. Bercovici, 1998, Interpolation with splines in tension: a Green's function approach,
Math. Geol., 30, 77-93.
Wessel, P., and J. M. Becker, 2008, Interpolation using a generalized Green's function for a spherical
surface spline in tension, Geophys. J. Int, 174, 21-28.
Wessel, P., 2009, A general-purpose Green's function interpolator, Computers & Geosciences, 35,
1247-1254, doi:10.1016/j.cageo.2008.08.012.
SEE ALSO
gmt, gmtmath, nearneighbor, psxy, sphtriangulate, surface, triangulate, xyz2grd
COPYRIGHT
2015, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe